Parabolic Pdes Explicit Method Homework Help


Explicit methods calculate the state of a system at a later time from the state of the system at the current time, implicit methods find a solution by solving an equation involving both the current state of the system and the later one. Mathematically, if is the current system state and is the state at the later time ( is a small time step), for an explicit method

 

Where for an implicit method one solves an equation

 

to find

It is clear that implicit methods require an extra computation (solving the above equation), they can be much harder to implement. The implicit methods are used because many problems arising in practice are stiff, to which the use of an explicit method requires impractically small time steps to keep the error in the result bounded (see numerical stability). To such problems, to achieve given accuracy, this takes much less computational time to use an implicit method with larger time steps, which even taking into account that one needs to solve an equation of the form (1) at each time step. It said, whether one should use an explicit or implicit method depends upon the problem to be solved.

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